In the continuous-time setting, a singular theory as required by (I) was already present in [1]}, later refined and clarified for fast-slow systems in standard form (REF ) in e.g. [2]}, [3]}, [4]}, [5]}, [6]}, and finally for the more general class (REF ) in [7]} and related works, e.g. [8]}, [9]}, [10]}, [11]}, [12]}, [13]}. This theory provides a mathematical framework for identifying and analysing the geometry and dynamics of the so-called layer (or fast subsystem) and reduced (or slow subsystem) problems, obtained after taking \(\varepsilon \rightarrow 0\)  in (REF ) on the fast and slow time-scales \(t\)  and \(\tau = \varepsilon t\)  respectively. A typical analysis proceeds by constructing `singular orbits', sometimes also called candidate orbits, by a geometric concatenation of fast trajectory segments of the layer problem on \(\mathbb {R}^n \setminus S\)  and slow trajectory segments of the reduced problem on \(S\) .
